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Introductory Measure Theory
Seminar 21:00:00 ChanServ changed the topic of #mathematics to: SEMINAR IN PROGRESS: If you want to ask a question say "!" and wait to be called 21:04:22 somiaj: so anyways we'll see how this goes. First off just a bit on prereqs. I plan to be abstract and general and not relay to heavly on any paticular example, but some understanding of the topology of R^n (specifically the reals R) and anylsis is helpful. 21:05:19 somiaj: So to start, measure is a generalization of volume. For example in the reals R, we can give intervals a length. 21:05:58 somiaj: We can say the length of the closed interval a,b is b-a (a 0,infity 21:38:32 somiaj: here 0,infity is the extended non-negative reals, we want to include infity, for example we say the measure of the reals is infinite, or the length of the real number line is infinite. 21:39:05 somiaj: The major property we want this function to follow is countably additive. First let's start with the finite case 21:39:29 somiaj: we say the function mu is finitely sub-additive if 21:40:00 somiaj: mu( union_{k-1}^n A_n ) <= sum_{k=1}^n mu(A_n) 21:40:28 somiaj: i.e. the measure of a union of sets is less than the sum of the measure of the sets itself. 21:40:50 somiaj: we then say the function mu is finitely additive if mu is finitely sub-additive and 21:41:31 somiaj: mu( union_{k=1}^n A_n ) = sum_{k=1}^n mu(A_n) where {A_n} is a disjoint collection of subsets of Omega. 21:42:15 somiaj: i.e. if we can write a set as the disjoint union of known sets then the measure of the full set is just the sum of the measures of the disjoint parts. 21:43:53 somiaj: Now of course we want this same idea to apply to any measureable set, so we can extend this and we acually want mu: Sigma -> 0,infity to be countaably additive. So let {A_n} be a countable collection of subsets 21:44:22 somiaj: then we want mu( union_n A_n ) <= sum_n mu(A_n) and equality to occur if {A_n} is a disjoint collection. 21:45:23 somiaj: Thus we have our full space, Omega, Our collection of measurable sets, Sigma, and our measure function, mu. All with properties of how we want measure to work. 21:45:53 somiaj: Here is a good place to add a theorem with some work, just to see how we acually work within our sigma-field. 21:46:10 somiaj: I'll just finish this by proving some basic results from my definitions. 21:46:20 somiaj: 1) mu(emptyset) = 0. 21:46:51 somiaj: Note emptyset = emptyset union emptyset, and this is a disjoint union since the emptyset is disjoint form itself 21:47:22 somiaj: thus mu(emptyset) = mu(emptyset union emptyset) = mu(emptyset) + mu(emptyset) we have finite additivity 21:47:40 somiaj: and for the real numbers we know that if c = c + c = 2c, then c=0. Thus mu(emptyset) = 0. 21:47:47 kommodore: ! 21:48:00 somiaj: yes 21:48:19 kommodore: are you rejecting the possibility mu(A)=infinity for all A? 21:49:23 somiaj: ahh yes, I belive so. most books define mu, such that mu(emptyset)=0, as part of its definition. 21:49:40 kommodore: ok 21:50:09 somiaj: Though I guess you could live in a space where mu(A) = infinity, but there wouldn't be much of interest if everything had infinite volume. 21:50:36 somiaj: here, look at 2) If A subset B, then mu(A) <= mu(B). 21:51:36 somiaj: proof: B = A union (B-A) since A subset B, and this is a disjoint union. 21:52:24 somiaj: Thus mu(B) = mu( A union (B-A) ) = mu(A) + mu(B-A) >= mu(A) by definition mu(B-A) >= 0 21:54:43 somiaj: so since emptyset subset A for all sets A, we have mu(emptyset) <= mu(A) for all A, so that in order for 1) to hold, ie mu(emptyset)=0, all we need is one set to have finite measure. (cause as kommodore pointed out having all sets of infinite measure would break 1) 21:55:49 somiaj: 3) if mu(A intersect B) < infity then mu(A union B) = mu(A) + mu(B) - mu(A intersect B) 21:57:27 somiaj: to see this, note that B = (B-A) disjoint-union (A intersect B) 21:58:36 somiaj: thus mu(B) = mu(B-A) + mu(A intersect B). Since mu(A interset B) < infity, we have mu(B-A) = mu(B) - mu(A intersect B) 21:59:34 somiaj: finally A union B = A disjoint-union (B-A), so mu(A union B) = mu(A) + mu(B-A) = mu(A) + mu(B) - mu(A intersect B) 22:01:33 somiaj: 4) if A subest B and mu(A) < infity, then mu(B-A) = mu(B) - mu(A). This follows from the subresult in part 5), mu(B-A) = mu(B) - mu(A intersect B) which holds if mu(A intesect B) < infity. And since A subset B, A intersect B = A. Thus we have the result. 22:02:50 somiaj: so again the basic construction is start with a full space Omega, from there define a collection of mesurable sets, or a sigma-field. On the collection of measurable sets define a non-negtive function mu: Sigma -> 0,infity 22:04:19 somiaj: I think of the definitions as being that we can do countable operations and want to preserve our mesure, so thus we want the function mu to be countabally additive. 22:04:48 somiaj: Looks like I'm out of time, but from here one can then take this basic construction and try to build the lebesgue measure. 22:05:15 somiaj: though building the lebesgue measure takes a lot of work to show that the function we create obeys all the properties I have listed. 22:05:27 somiaj: any questions so far (almost feel as if I'm talking to myself) 22:05:49 _llll_: what sort of morphisms of measurable space do people use? 22:07:29 somiaj: I can't think of any, most of the studying I have done of measurable spaces quickly developes to the general lebesgue intergral and then creating the L_p(Omega) spaces. 22:09:20 somiaj: The two main paths from here are going into the lebesgue Measure, or looking at R^n, and generalizing volume there. In these cases mu(R^n) = infity, so we have an infite measure space. 22:09:55 pyninja: Thanks somiaj, some of it went over my head but it was still interesting. 22:10:03 somiaj: Though in the case of R^n we can define it to be a sigma-finite measure. 22:10:03 _llll_: are you going to follow one of these paths in a follow-up seminar next week? 22:11:38 somiaj: so we say mu is sigm-finite if there exists a sequence ( Omega_1 subset Omega_2 subset Omega_3 subset .... ) and Omega = union_n Omega_n Such that mu(Omega_n) < infity for all n 22:12:05 somiaj: If mu(Omega) < infity we call mu a finite-measure, the most common one of these is when mu(Omega) = 1. 22:12:31 somiaj: if mu(Omega) = 1, then (Omega, Sigma, mu) form a probability space, which is the second main path you can go from here) 22:13:16 somiaj: would people be intersted, I could continue though my notes to develope outter measure, and then the lebesgue measure and if we have time look at the cantor set. maybe get our hands on some examples that are less definitions and abstract. 22:13:37 _llll_: id be interested 22:14:20 * ichor would be interested too. 22:14:29 |Steve|: And I. 22:15:07 ~DWarrior-: I'd definitely be interested next week 22:15:19 burned: I would be too 22:16:31 somiaj: we should try to limit these to an hour. 22:19:49 somiaj: though one note above, on analogy that I liked was thinking back to simple high-school math problems where you want to find the area of an object by spliting it up into pieces you know the area of (rectangles, triangles, circles) and then adding the desired areas to get the result 22:20:23 somiaj: That is kinda what the sigma-field and countabally additive function do, but our shapes can become far more bizar. 22:23:53 ChanServ changed the topic of #mathematics to: NEXT SEMINAR: The Lebesgue Measure by somiaj Sunday 27 July 20:00 UTC | Transcript of last seminar: http://www.freenode-math.com/index.php/Introductory_Measure_Theory | Future Seminars: http://www.freenode-math.com/index.php/Seminars Category:Seminar Category:Measure Theory